Lecture 14 of 'Scientific Computing' (wi4201)

• The following subjects are discussed:
  
  
  • Iterative methods for eigenvalue problems
  • Applications where eigenvalues are important
  • Definition of eigenvalues and eigenvectors
  • Only iterative methods can be used to approximate eigenvalues and eigenvectors
  • Power method and Krylov methods: Lanczos and Arnoldi
  • Critical force to bend a beam, connection with eigenvalue computation
  • Power method to approximate the in absolute value largest eigenvalue
  • Example of the Power method
  • Amount of work and memory
  • Linear convergence of the Power method
  • Proof of the convergence result
  • Starting vector and stopping criteria
  • Shifted power method
  • Inverse power method
  • Shift and Invert Power method
    
    
• Material is described in pages 129-132 of the lecture notes. Recommended exercises: 8.6.7 and 8.6.9